The failure of rock-like, quasi-brittle materials is influenced by the development of an intrinsic process zone in the form of a localized region of microcracking. In particular, the process zone has a fundamental importance for defining the structural or system behavior in terms of the post-peak instability, a qualitative size effect, and the maximum stress or material strength, a quantitative size effect. For geometrically similar beams of different sizes, this paper presents experimental evidence from locations of acoustic emissions of the process-zone development, at maximum stress, in terms of shape and size. The experiments suggest that a notch effect developed in the specimens, with assumed uniform tractions being transmitted between the two boundaries of the notch; the radius of curvature of the notch was taken to be one-half the width of the intrinsic process zone. The failure criterion was that at peak load the stress at the notch tip reached the theoretical tensile strength, the so-called characteristic strength of the material. It is shown that the quantitative size effect can be explained by extending the classical Neuber results on stress concentrations around notches through an account of the intrinsic process zone and its cohesive interaction. A notable outcome of the analysis is that two competing factors define the nominal strength of a quasi-brittle material: the positive contribution of the process zone and the competing aspects of the undamaged volume, that is, the size.